Differentially Closed Fields

Differentially closed fields are mathematical structures that contain solutions to differential equations, similar to how algebraically closed fields contain solutions to polynomial equations. They are "complete" in the sense that any differential equation that can possibly have a solution within the structure does indeed have one there. This concept helps mathematicians understand how differential equations behave in an abstract, algebraic setting, ensuring the field is rich enough to include solutions whenever they should exist, leading to a well-behaved theory of differential equations within this mathematical framework.